Second and third virial coefficients for polyisobutylene in cyclohexane

Excluded-Volume Effects in Star Polymer Solutions: Four-Arm Star Polystyrene in Benzene. Mitsuhiro Okumoto, Yo Nakamura, Takashi Norisuye, and Akio ...
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Macromolecules 1992,25, 1134-1138

1134

Second and Third Virial Coefficients for Polyisobutylene in Cyclohexane Yo Nakamura, Kazutomo Akasaka, Kenichi Katayama, Takashi Norisuye,' and Akio Teramoto Department of Macromolecular Science, Osaka University, Toyonaka, Osaka 560, Japan Received August 13,1991; Revised Manuscript Received October 28,1991

ABSTRACT Light scattering measurements were made on cyclohexane solutions of 10 polyisobutylene (PIB) fractions ranging in weight-average molecular weight M , from 2.6 X l(r to 1.4 X 10' at 25 "C to determine the molecular weight dependence of the second virial coefficientAz, the third virial coefficientAs, and the z-average radius of gyration (s2),1/2. Zero-shear intrinsic viscosities were also determined as a respectively,and the factor function of M,. For M , higher than 7 X l(r,A3 and ( Sa), vary as M,o.6 and Mw1.19, g defined by Aa/AzzM, increases with increasing M,. The interpenetration function \k and g plotted against the cube of the radius expansion factor, a:, are almost superimposed on the respective relations known for polystyrene in benzene. As a : increases, \k gradually decreases to an asymptotic value of 0.224.24, while g increases to 0.45-0.50. It is found that no available theory can explain the molecular weight dependence of A3 for the two typical flexible polymers, though the g w a? relation predicted early by Stockmayer and Casassa happens to come close to the experimental curve for as3> 2.

Introduction In our previous work,lS2it was found from light scattering measurements that the third virial coefficient A3 of polystyrene in benzene varies in proportion to M,o.6 for M, higher than 2 X 105 and that, in the same M, region, the reduced thud virial coefficient g defined by A3/A22M, increases monotonically with increasing M,. Here, as usual, M, and A2 denote the weight-average molecular weight and the second virial coefficient, respectively. Apparently, similar studies on other polymer + solvent systems are needed to see whether these findings are common to long flexible chains in good solvents. Thus, in the present work, with polyisobutylene (PIB) in cyclohexane at 25 "Cchosen as sucha system, light scattering experiments were performed to determine A2 and A3 as functions of M,. The PIB fractions prepared in this study covered a broad range of M, from 2.6 X 104 to 1.4 X lo7. The M, value ~ -far ~ reported of 1.4 X lo7was the largest among t h ~ s e so for this polymer. Thus we also determined z-average mean-square radii of gyration ( S2)zand zero-shear intrinsic viscosities [VI for the PIB fractions in cyclohexane, hoping to establish their asymtotic behavior at high molecular weight. In this connection, we note that, although PIB is known as a typical flexible polymer, published (S2),data in the solvent are yet limited to those reported early by Matsumoto et The results obtained for A2, As, (S2),, and [TI are compared with literature data for the same system4s6Jor the polystyrene + benzene system,172and the chain length dependence of A2 and A3 is discussed in relation to t h e two-parameter theory. Experimental Section Samples. Three commercial PIB samples were used. Two of them (higher molecular weight samples) were Enjay's Vistanex L-80 and L-300 stored in our laboratory, and the other (lower molecular weight sample), designated below as S, was obtained from Scientific Polymer Products, Inc. From these samples 10 fractions were prepared by fractionation as described below. Samples L-80 and L-300 were each separated into seven parta using the 8 column elution method with benzene as the solvent. The experimental procedures were essentially the same as those employed by Matsumoto et al.' The middle parta were fractionated again by the same method to obtain 3-5 fractions from Oo24-9297/92/2225-1134$03.OO/O

each. Appropriate fractions, designated A-22, A-42, A-62 (from L-80). P-32, P-53, and P-62 (from L-300), were chosenfrom many fractions extracted in this way. The f i t three fractions were further fractionally precipitated with benzene as the solvent and methanol as the precipitant to remove lower and higher molecular weight portions. The fractionsthus obtained were designated A-22B4,A-42B3, and A-62B1and were used for the present etudy, along with fractions P-32, P-53, and P-62. These six fractions were reprecipitated from benzene solutionsinto acetoneand dried in vacuo for about 1 week. Sample S was fractionated by the column method with a benzene-methanol mixture instead of pure benzeneas the eluent; the composition of methanol in the mixture was adjusted so that the solution became turbid at about 23 "C. However, a larger portion of the polymer tended to elute at a lower temperature, and the column method seemed less effective for a lower molecular weight sample. Thus, after this method had been repeated, the fractions obtained were subjected to fractional precipitation in benzene-methanol mixtures. In this way, four fractions, designated below as S-111B, S-l12B, S-114B, and S14B, were prepared. They were freeze-dried from cyclohexane solutions after being reprecipitated from benzene solutions into methanol. Light Scattering. Intensities of light scattered from cyclohexane solutions of PIB were measured at 25 "C on a Fica 50 light scattering photometer in an angular range from 22.5 or 30 to 150"; for the two highest molecular weight fractions P-53 and P-62, the measurement was made from 15 to 41" at intervals of 2.0 or 2.5O. Vertically polarized incident light of 436-nm wavelength was used for seven lower molecular weight fractions and that of 546 nm for the others. Benzene at 25 "C was used to calibrate the apparatus. Its Rayleigh ratio was taken to be 46.5 x 1O-g cm-l for 436 nm and 16.1 x lo* cm-l for 546 nm,Band its depolarization ratio was determined to be 0.41 and 0.40 for 436 and 546 nm, respectively, by the method of Rubingh and Yu.10 Test solutions were prepared as follows. First, a given PIB fraction was mixed with cyclohexane by stirring for 2-7 days in an air bath thermostated at 30 "C;for the two highest molecular weight fractions, very gentle stirring was applied to prevent the polymer from degradation. The solution was then diluted with the solvent to a series of polymer concentrations. In this way, seven solutions of different concentrations were prepared for each fraction. They were optically clarified by the method established in this laboratory (see, for example, ref 11). The cyclohexaneused was fractionally distilled after being refluxed over sodium for more than 4 h. The polymer mass concentration c in each solution was calculated from the solute weight fraction, with the solution 0 1992 American Chemical Society

Second and Third Virial Coefficients for PIB 1136

Macromolecules, Vol. 25, No. 3, 1992

CI+C2/

1

2.2

..-

0

0.5

0

sin2(8/2)

IO'^^.^ I

1.S

2

Figure 2. Plots of S(cl,c2) va c1 + c2 for PIB fractions in cyclohexane at 25 "C. S(cl,c2) data for pairs of neighboring c1 and c2 in a series of polymer concentrations are omitted, since they were not very accurate (see eq 1).

Figure 1. Angular dependence of the scattering intensity for PIB fraction A-22B4 in cyclohexane at 25 "C. The polymer concentrations are 9.755 X lo-L,8.301 X l p , 6.954 X lo-', 5.523 X 1o-L,4.170 X lo4, 2.803 X lo-L, and 1.440 X 1o-L g cm-3from top to bottom. densityapproximated by the solvent density. Thie approximation introduced errors of less than 0.5% in the values of A2 and A3 for all fractionsstudied. The highest c studiedby light scattering was about 2 X 1V2g ~ m (see - ~Figure 3). Excess refractive indices for PIB in cyclohexane at 25 "C were measured at concentrationsbelow 2 x g cm4 using amodified Schulz-Cantow type differential refractometer and a doublebeam differentialrefractometer (otsukaElectronics,DRM-1020). The results for the specific refractiveindex increment were 0.0991 and 0.0966 cm3g l at 436 and 546 nm, respectively, independent of c in the range examined. These values agreed closely with 0.0987 cm3gl (at 436 nm) and 0.0961 cm3g l (at 546 nm) obtained by Tong et al.12 Matsumoto et al.' reported a slightly larger value of 0.1008 cm3 g l at 436 nm. Viscometry. Zero-shear intrinsic viscosities for the three highest molecular weight fractions in cyclohexane at 25 "C were determined by a low-shear four-bulbcapillaryviscometer of the Ubbelohde type. For the other fractions,a conventionalcapillary viscometer was used.

0.5

c/i~-~gcm'~ 0

145

e

-

- -

I

,

2

1 .

"

I

P-62

2

1- -

-- -_ - _- - -

9.0 l5 8.6 5.0

e

.,

4.6 I 0

I

e

.,

A-6281

A-4283

-

-. , I

5

A-2284

& I

10

c~io~gcm'~

s =

-

I

ul

3

-

I I

c/ 1

o

5

0

1

10

Results Data Analysis. Figure 1 illustrates the angular dependence of (Kc/Ro)'f2 for fraction A-22B4 in cyclohexane a t 25 "C. Here, K is the optical constant and Re is the reduced scattering intensity at scattering angle 0. From the zero-angle values of Kc/Re (i.e., Kc/Ro) obtained by extrapolation for the fraction and those similarly obtained for other fractions, plots of S(c1,cz) vs c1 + c2 were constructed according to the expression of Bawn et al.I3 (originally derived for osmotic pressure) S(C,,C~) I [(Kc/RJc=c2- (KC/&)~=~,I/(C~ - ~ 1 =) 2-42 +

+

3A,(ci + c2) 4A4(cl2+ c1c2 + c:) + ... (1) where A4 denotes the fourth virial coefficient and c1 and c2 denote different c values. The resulting Bawn plots are shown in Figure 2. The plotted points for each fraction fall on a straight line, allowing separate determination of A2 and A3. We note that when contributions from A4 to S(c1,cz) are not negligible, a Bawn plot should bend or even show ~p1its.l~

2

C/1

Figure 3. Plots of Ma vs c for PIB fractionsin cyclohexane at 25 O C . is definea by eq 2. With A2 and A3 thus evaluated, we calculated Mapp defined by

Mapp= [(Kc/RO) - 2A2~- 3A3~'I-l (2) as a function of c for each fraction. Figure 3 shows that the plots of Mapp vs c constructed are essentially horizontal for any fractions and thus permit unambiguous extrapolation of Mapp to c = 0. The intercepts give M, for the respective fractions. The values of M,, A2, and A3 determined are s u m m a r i d in Table I, along with those of (s2),1/2 obtained by Berry's square-root plotlS (for fraction S-l14B, (s2),1/2 was

Macromolecules, Vol. 25, No. 3, 1992

1136 Nakamura et al. Table I Results from Light Scattering Measurements on PIB Fractions in Cyclohexane at 26 O C

MwX A2 X l@, A3 X 102, (S2)z'/2, fraction 10-6 mol g2 cm3 mol 8-3 cm6 nm Mw/Mna 0.88 0.261 10.6 S-111B 1.i6 1.08 0.429 9.55 S-112B 0.777 8.40 S-114B 1.50 11.6 1.10 2.6 18.7 1.09 1.81 7.05 S-14B 4.7 33.4 1.10 4.88 5.30 A-22B4 8.78 4.55 A-42B3 6.3 47.6 1.09 9.2 66.8 3.85 A-62B1 15.8 P-32 14.0 107 33.9 3.13 27 177 P-53 78.3 2.68 42 252 P-62 141 2.54

-E

10' :

;?

. E 9 lo2 I

1' 0

,

1

1

1

,

1

1

1

1

1o5

I

1os

,

1

1

,

1

1

,

1

I

I

,

1' 0

MW

From gel permeation chromatography.

Figure 5. Molecular weight dependence of A3 for PIB in cyclohexane at 25 "C. The slope of the straight line is 0.6.

i

1o4

loe

1o5

1o7

0 '

MW

Figure 4. Molecular weight dependence of A2 for PIB in cyclohexane: (0)this work; (@) Mataumoto et d,;4(A)Fetters et d.7

'

Io4

* * , , ) ( , '

'

1

1

1

*

1

*

1

1

o6

1o5

1

'

, l l l . l * '

1o7

'

, I

MW

Figure 6. Molecular weight dependence of g for PIB in cyclohexane at 25 OC.

evaluated by the method described elsewhere"% The last column of this table presents ratios of the weight-average to number-average molecular weight, M,/M,, estimated from gel permeation chromatography in chloroform. Molecular Weight Dependence of Virial Coefficients. The molecular weight dependence of A2 for PIB in cyclohexane is illustrated in double-logarithmic form in Figure 4, in which the data of Matsumoto et aL4(at 25 "C)and Fetters et aL7 (at 23 "C)are also shown for comparison. The solid curve fitting our data points is almost linear with a slope of -0.24 for M , below 2 X lo6 and slightly bends upward for higher M,. The indicated dashed line, which is almost tangential to this curve for M , > 2 X 106, has a slope of -0.20. This exponent is the asymptotic value predicted by two-parameter theories.17 The data of Matsumoto et al. appear slightly above the solid line but give the exponent -0.23 close to our value of -0.24. On the other hand, those of Fetters et al. deviate downward from the line for M , below 4 X lo5,yielding an exponent of about -0.20. Figure 5 shows that A3 for PIB in cyclohexane increases almost in proportion to M,o.6 over the entire range of molecular weight studied. This exponent of 0.6 agrees with that obtained previously112for polystyrene in benzene for Mwabove 2.5 X 105. We note that it is the asymptotic value expected from the two-parameter theory. The values of g calculated from the data of A2, AS,and M , in Table I are plotted against log M, in Figure 6. It can be seen that g stays at about 0.28 up to M, 2 X 105 and then gradually increases to 0.45-0.50 with increasing M,. This behavior of g is also similar to what we observed previously2for polystyrene in benzene in the region of M , above lo5. A quantitative comparison between g of the two polymers is deferred to the Discussion section. Molecular Weight Dependence of (#),and [TJ].The in Table I are plotted double-logarithmically values of ( P),

-

10"

MW

Figure 7. Molecular weight dependence of (P), for PIB in cyclohexane at 25 O C : ( 0 )this work; (@) Mataumoto et al.' against M , in Figure 7, together with the data of Matsumoto et al.4 Our data points are fitted by a straight line with slope 1.19 (fO.O1) throughout the entire range of M , indicated. This slope is to be expected for long flexible chains with large excluded volume.'* It perfectly agrees with the exponent determined for polystyrene in benzene by Miyaki et al." from their own (P),data and those of Yamamoto et al.19 and Fukuda et aLz0covering a very wide range of M, from 1.8 X lo5 to 5.7 X lo7. The PIB data of Matsumoto et al. appear systematically below o w , but the slope 1.17 reported by them does not differ much from 1.19. The values of 173 and k' (Huggins' constant) determined for all PIB fractions in cyclohexane at 25 "Care presented in Table 11, and these [VI data are compared with those

Second and Third Virial Coefficients for PIB 1137

Macromolecules, Vol. 25, No. 3, 1992 Table I1 Results from Viscosity Measurements on PIB Fractions in Cyclohexane at 26 OC fraction [ q ] x W, cm3gl k’ 0.39 0.276 S-111B 0.36 0.379 S-112B 0.36 0.592 S-114B 0.36 1.07 S-14B 0.38 2.32 A-22B4 0.38 3.56 A-42B3 0.38 5.32 A-62B1 0.35 9.46 P-32 0.38 18.3 P-53 0.35 27.4 P-62

0.3

0.2

* 0.1

J I

0

I

1

I

I

I

I

I

I

5

I

I

10

“2 Figure 9. Plota of vs a2 for PIB in cyclohexane (0,present data; 0, ref 4) and polystyrene in benzene (0,refs 11and 23; 9, ref 19; &, ref 20). Dot-dash line, Barrett’s theory.”

r

0

-

*I

. I,, I

C

I

The dot-dash line in Figure 9 represents Barrett’s theory24

-

-

+

a

‘lllal’

’ ’

l ’ a l l l ’

a

‘‘llall‘

+

9 = (z/a:)(l 14.32 57.3z2)+’.’ (4) combined with the Domb-Barrett e x p r e s s i ~ nfor ~ ~a8:

A

0.1

‘

[ + 102 +

a : = 1

of Matsumoto et al.,4 Gundert and Wolf: and Fetters et aL7in Figure 8. As a whole, the four data sets agree with one another. The curve drawn to fit our data points has a slope of 0.75 for M , > 7 X 104,which is also in substantial agreement with 0.76 reported by Matsumoto et al. and 0.74 by Fetters et al. Discussion Interpenetration Function. The second virial coefficient is usually discussed in terms of the interpenetration function 9 defined by17 (3) where NAis Avogadro’s constant. We calculated 9 from the data of M,, A2, and (S2),1/2in Table I. The unfilled circles in Figure 9 show the resulting il? values plotted against as3,the cube of the radius expansion factor (as2= (S2),/(S2)oZ, with (S2)o2being the value of (S2),in the 8 state). We have evaluated21 as using the experimental relation ( S 2 ) ~ , / M=W9.52 X cm2 in isoamyl isovalerate, reported by Matsumoto et aL4 The graph includes the 9 data of these authors for PIB in cyclohexane (the half-filled circles) and also those of Yamamoto et al.,19 Fukuda et al.,2O and Miyaki et al.11*23for polystyrene in benzene (the filled circles with or without pip). The unfilled and filled circles are fitted by the single solid curve indicated, being consistent with the prediction from the two-parameter theory that 9should be a universal function of as.The asymptotic value of 9 for large asappears to lie between 0.22 and 0.24. The half-filled circles deviate upward from the solid curve. This deviation reflects the small differences in both A2 and ( S2), between our data and Matsumoto’s, observed in Figures 4 and 7.

z2

+ 8~3

12 z 3 1 2/15

[0.933

+

0.067 exp(-0.85~- 1.39z2)1 (5) Here, z is the familiar excluded-volume parameter. Satisfactory agreement is observed between the dot-dash and solid curves for as3 greater than 6. Equation 4, when a? = 1 . 5 3 ~ ~ / ~ combined with the asymptotic relation11*26 derived from computer simulation data, yields an asymp totic 3 of 0.235,18124which is close to the above estimate (0.22-0.24) for the two typical flexible polymers. However, with a decrease in as3, the theoretical curve declines, while the experimental one rises. A similar opposite trend of 9 at relatively large ashas already been as well as for observed for poly(~-j3-hydroxybutyrate)~~ p ~ l y s t y r e n e .The ~ ~ present PIB data provide additional evidence for this serious discrepancy. As discussed by Fujita and Norisuye,28the observed opposite trend may be ascribed primarily to the fact that eq 4 with eq 5 predicte a molecular weight dependence of A2 weaker than that represented by A2 a while the experimental A2 for PIB (that for polystyrene as well) has a stronger dependence (see Figure 4). We should add that, in contrast to the dot-dash line sharply declining to zero, il? for low molecular weight polystyrene in t o l ~ e n rises e ~ ~first ~ ~grad~ ually and then sharply as asapproaches unity. Reduced Third Virial Coefficient. We have found that the molecular weight dependence Of A3 andg for PIB in cyclohexane is quite similar to that for polystyrene in benzene. For a quantitative comparison, the values of g for the two systems are plotted against a83in Figure 10. For fractions whose (S2), data are unavailable, we have estimated a,by extrapolating the empirical relations as2 = 0.206M,0.19 for PIB and as2= 0.167M,0.1s for polystyrene;Il~ the ~ ~former was obtained from our data in Figure 7 and the (S2)o2data of Matsumoto et al. Because of the extrapolation, the abscissa values below 2 are inaccurate, but their accuracy is immaterial in the discussion below. The plotted points for the two polymers approximately fall on a single curve, at least, in the region of as3above 2, as indicated by a solid line. This finding is consistent

Macromolecules, Vol. 25,No. 3, 1992

1138 Nakamura et al.

in cyclohexane and polystyrene in benzene. Even qualitatively, there exists a serious discrepancy in g between our experiments and the two-parameter theory when an is approached to unity by lowering the molecular weight in a given good solvent. The discrepancy is similar to what is known for 9,indicating that the current twoparameter theory for A3 overlooks something important, as is the case with that for A2. References and Notes I 1

I

I

I

5

I

I

I

I

J

10

“P Figure 10. Plots of g VB a: for PIB in cyclohexane ( 0 )and polystyrene in benzene (4,ref 1;0,ref 2). Dot-dash line, Stockmayer and Caeassa’e theory.” with the requirement of the two-parameter theory. The solid curve rises with increasing asand appears to level off at 0.45-0.50. This leveling-off value of g is close to 0.44, evaluated by Knoll et al.31and des Cloizeaux and Noda,32 for good solvent systems on the basis of the first-order t expansion in the renormalizationgroup method. However, this agreement should be accepted with reservation until a higher order calculation becomes available. Further, as notad previ~usly,~J~ all the renormalization group calso~far culation~ ~ made ~ ~ ~ fail to explain why g for a good solvent system depends on M. On the other hand, the two-parameter theories worked out early by Stockmayer and Casassa,34K ~ y a m a ?and ~ Y a m a k a ~ aall~ predict ~ g to increase monotonically with increasing M. The dot-dash line in Figure 10 represents the g vs ag3 relation predided by Stockmayer and Casassa, who combined their smoothed-density theory for g (as a with the original Flory equation3’for as, function of dag3) i.e., a,S - as3= 2.602. This line comes close to the solid curve for as3> 2. However, as ae approaches unity, it sharply declines to zero, while the experimentalg stays at about 0.3 or even goes up after passing through a shallow minimum. This sharp contrast is similar to what has been observed for 9.30 As was shown by Miyaki and Fujita!E the original Flory equation used by Stockmayer and Casassa is a good approximation to either PIB in cyclohexaneor polystyrene in benzene for an3> 2, so that the observed agreement between the solid and dot-dash curves in Figure 10 may be taken as that ing itself. However, this agreement must be due to a compensation of the errors in both A2 and A3 that the Stockmayer-Casassa smoothed-density theory involves, because the Flory-Krigbaum the0ry3~for A2 based on the game smoothed-densitymodel fails to describe A2 for flexiblepolymers. In other words, the StockmayerCasassa theory should be invalid for AS,though it almost quantitatively explains the ratios of As to A22Mwfor the two typical flexible polymers in the region of as3above 2. Koyama’s the0ry3~is essentially the same as the Stockmayer-Cmassa theory, while Yamakawa’s theory36based on the differential-equation approach gives g values that are too large. In conclusion,none of the available theories can explain the observed molecular weight dependence of A3 for PIB

(1) Sato, T.; Norisuye, T.;Fujita, H.J. Polym. Sci., Part B: Polym. Phys. 1987,25, 1. (2) Nakamura, Y.; Norisuye, T.; Teramoto, A. J. Polym. Sci., Part B: Polym. Phys. 1991, 29, 153. (3) See references cited in the papers of Mataumoto et al.4 and

Fetters et al.’ for early data. (4) Mataumoto, T.; Nishioka, N.; Fujita, H. J.Polym. Sci., Polym. Phys. Ed. 1972, 10, 23. (5) Gundert, F.; Wolf, B. A. Makromol. Chem. 1986,187,2969. (6) Gundert, F.; Wolf, B. A. J. Chem. Phys. 1987,87,6156. (7) Fetters, L. J.; Hadjichristidis, N.; Lindner, J. S.; Mays, J. W.; Wilaon, W. W. Macromolecules 1991,24, 3127. (8) Abe, F.; Einaga, Y.; Yamakawa, H. Macromolecules 1991,24, 4423. (9) Deblie, Gj.; Vavra, J. Croat. Chem. Acta 1966, 38, 35. (10) Rubingh, D. N.; Yu, H. Macromolecules 1976, 9,681. (11) Miyaki, Y.; Einaga, Y.; Fujita, H. Macromolecules 1978, 11, 1180. (12) Tong, Z.; Einaga, Y.; Kitagawa, T.; Fujita, H. Macromolecules 1989,22,450. (13) Bawn, C. E. H.; Freeman, R. F. J.; Kamaliddin, A. R. Trans. Faraday SOC.1960,46,862. (14) Norisuye, T.; Fujita, H. Chemtracts, in press. (15) Berry, G. C. J. Chem. Phys. 1966,44,4550. (16) Kitagawa,T.; Sadanobu, J.;Norisuye,T. Macromolecules 1990, 23, 602. (17) Yamakawa, H. Modern Theory of Polymer Solutions; Harper & Row: New York, 1971. (18) Fujita, H. Polymer Solutions; Elsevier: Amsterdam, Holland,

1990.

(19) Yamamoto,A.; Fujii, M.; Tanaka, G.; Yamakawa, H. Polym. J. 1971, 2, 799. (20) Fukuda, M.; Fukutomi, M.; Kato, Y.; Hashimoto, T. J. Polym. Sci., Polym. Phys. Ed. 1974, 12, 871. (21) Near the completion of the present study, reference was made to the work of Konishi et al.,22who obtained ( Sz)&/M, for four PIB fractions with M, = 4.22 X 105-1.76 X 106 in isoamyl is-

ovalerate. Use of their data gives a,3values that are only about 5 5% smaller thanthose estimated from Mataumotoet al.’s ( P ) o r data. (22) Koniehi,T.; Yoshizaki,T.;Yamakawa,H.Macromolecules1991, 24,5614. (23) (24) (25) (26)

Miyaki, Y. Ph.D. Thesis, Osaka University, 1981. Barrett, A. J. Macromolecules 1986, 18, 196. Domb, C.; Barrett, A. J. Polymer 1976, 17, 179. Lax, M.; Barrett, A. J.; Domb, C. J.Phys. A: Math. Gen. 1978,

11, 361. (27) Miyaki, Y.; Einaga, Y.; Hirosye, T.; Fujita, H. Macromolecules 1977,10, 1356. (28) Fujita, H.; Norisuye, T. Macromolecules 1986,18, 1637. (29) Huber, K.; Bantle, S.; Lutz, P.; Burchard, W. Macromolecules 1986, 18, 1461. (30) Huber, K.; Stockmayer, W. H. Macromolecules 1987,20,1400. (31) Knoll, A.; ScWer, L.; Witten, T. A. J.Phys. (Orsay,Fr.) 1981, 42,767. Schser, L. Macromolecules 1982, 15, 652. (32) des Cloizeaux, J.; Noda, I. Macromolecules 1982, 15, 1506. (33) Douglas, J. F.; Freed, K. F. Macromolecules 1986,18, 201. (34) Stockmayer, W. H.; Casassa, E. F. J. Chem. Phys. 1952, 20, 1560. (35) Koyama, R. J. Chem. Phys. 1957,27, 234. (36) Yamakawa, H. J. Chem. Phys. 1965,42, 1764. (37) Flory, P. J. J. Chem. Phys. 1949,17, 303. (38) Miyaki, Y.; Fujita, H. Macromolecules 1981, 14, 742. (39) Flory, P. J.; Krigbaum, W. R. J. Chem. Phys. 1950,18, 1086.